Van Quy (Optimization 68(4):753–771, 2018) established an extragradient–CQ algorithm for solving a class of bilevel split equilibrium problem. The step-size in the algorithm requires computation of a certain matrix norm which is costly. Moreover, bilevel problems often possess huge number of constraints and possibly require a robust and adaptive step-size algorithm to withstand failure in instances of real-world problems with large data sizes. In this paper, we propose a self-adaptive step-size extragradient–CQ algorithm for solving the same problem without prior knowledge of operator norm and provide a numerical example to demonstrate the effectiveness of our method.

Van Quy（优化68（4）：753–771，2018年）建立了一种超梯度CQ算法来解决一类双水平分裂平衡问题。该算法中的步长要求计算一定的矩阵范数，这很昂贵。而且，双层问题通常具有大量约束，并且可能需要鲁棒且自适应的步长算法来承受大数据量的现实问题的故障。在本文中，我们提出了一种自适应步长超梯度CQ算法，该算法可以解决相同问题而无需算子规范的先验知识，并提供了一个数值示例来证明我们方法的有效性。